What Is Calculus About?

Rating:
Summary: An intuitive examination of the concept of the derivative.
Review: Sawyer's text introduces the concept of the derivative to students who are preparing to take a calculus course. Since Sawyer assumes only a solid foundation in algebra and some experience with graphing functions, the text is accessible to the lay reader. He takes an inutitive approach; formal definitions and proofs are not provided.

Sawyer begins his study of the derivative by looking at velocity. He describes the difference between average and instanteous velocity. He shows that instanteous velocity, the derivative of the position function, can be obtained as the limit of the average velocities as the time interval approaches zero. Sawyer then defines the derivative as the slope of the tangent line to a curve at a point, provided it exists. He shows that this definition is equivalent to the first. To keep things simple, he confines his attention to polynomial functions and reciprocal powers of the variable.

Sawyer uses his results to show how the derivative can be used to describe acceleration, curvature, and the behavior of graphs. The final chapters describe, without going into detail, how (integral) calculus can be used to compute areas and volumes and how calculus can lead to nonintuitve results.

Sawyer employs many diagrams and well-chosen examples to show why the formulas that he derives are reasonable. However, I think that formal definitions and proofs would have made his arguments far more convincing. Sawyer stresses conceptual understanding of the derivative, but the exercises, for which answers are provided in the back of the text, consist mostly of repetitive applications of the formulas he derives in his exposition.

The material in this text is covered in good calculus texts, making this text superfluous for students taking calculus. It is best suited to students who want to understand the basic concepts of calculus without having to deal with the formal machinery of calculus.

Rating:
Summary: easy guide to difficult calculus
Review: When I was reading this book all along, I felt comfortable. I heard it as if my grandfather told an interesting story. When the story was over, I became a real university student. I had learned a vacant caculus, but I have known that the calculus 'is' our life.

Rating:
Summary: Mathematics of calculus, made simple. Really!
Review: Whenever I teach calculus, my emphasis is always on the fact that the basic ideas of calculus can be understood by almost anyone. While I am not always successful in proving this to the students, Sawyer certainly would be. His explanations of the basics of a derivative are the clearest, most understandable that I have ever seen. There are many diagrams, and each one has a specific purpose and they are well integrated into the textual explanations.
This is not a book that you could use to teach a college calculus course. The mandatory epsilons and deltas that form the backbone of basic calculus are mentioned only as an incidental. Sawyer sets out to explain the foundation ideas of calculus in terms of everyday occurrences and for that reason it is better suited to someone who is curious about calculus. However, it could be used as a supplemental text in the foundations of science, as calculus is used in all areas of change, which describes almost all of nature.
While the notation of mathematics is concise, abstract and often appears esoteric, many of the ideas expressed in that notation are quite easy to follow. In this book, Sawyer explains what calculus is all about in terms that anyone who understands motion can follow. There needs to be more people like him writing books like this.